---
tag: QV75TT6131
kind: statement
role: theorem
name: Cheeger–Gromoll splitting theorem
msc: [53C21, 53C20]
uses: [7CBT4SM83Y, 4GTNM0WHDH, NSYMQMHPDN, B4QR306D8G, R9YWQS15AA]
about: [4GTNM0WHDH]
sources:
  - url: https://en.wikipedia.org/wiki/Splitting_theorem
---

## Statement

Let $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] with nonnegative [[Ricci curvature|4GTNM0WHDH]].
Suppose $M$ contains a line, that is, a unit-speed [[geodesic|NSYMQMHPDN]] $\gamma \colon \R \to M$ with $d(\gamma(s), \gamma(t)) = \abs{s - t}$ for all $s, t \in \R$, where $d$ is the [[Riemannian distance|B4QR306D8G]].
Then $M$ is [[isometric|R9YWQS15AA]] to a Riemannian product $N \times \R$, where $N$ is a connected complete Riemannian manifold with nonnegative Ricci curvature.

## Remarks

The idea of the proof: let $b^{+}$ and $b^{-}$ be the Busemann functions of the two rays $t \mapsto \gamma(t)$ and $t \mapsto \gamma(-t)$, $t \ge 0$.
The Laplacian comparison theorem gives $\Delta b^{\pm} \le 0$ in a weak sense.
The triangle inequality gives $b^{+} + b^{-} \ge 0$, with equality on $\gamma$, so the strong maximum principle forces $b^{+} + b^{-} = 0$ and $b^{+}$ is harmonic and smooth.
The [[Bochner formula|DF7KYMAGTC]] applied to $b^{+}$, with $\abs{\nabla b^{+}} = 1$, gives that its Hessian vanishes, so $\nabla b^{+}$ is a parallel vector field and $M$ splits along its flow.

Example: the flat cylinder $S^1 \times \R$ contains the lines $\set{\theta} \times \R$ and splits with $N = S^1$.

The curvature hypothesis is needed: the hyperbolic plane contains lines but is not a product with a line.
The line is needed too: the paraboloid $z = x^2 + y^2$ has positive curvature and contains no line.

## Explanations

### L1 human

Suppose a space is nowhere negatively curved in the Ricci sense and contains one infinitely long shortest path.
Then the whole space is a product of a cross-section and that line, like a cylinder.

### L2 human

Toponogov proved the theorem earlier under the stronger hypothesis of nonnegative sectional curvature; Cheeger and Gromoll proved the Ricci version in 1971.
A main consequence: if $M$ is compact with $\operatorname{Ric} \ge 0$, then its universal cover is isometric to $N \times \R^k$ with $N$ compact.
It follows that the fundamental group of such $M$ contains a free abelian subgroup of finite index.
Versions of the theorem for limit spaces are central in the work of Cheeger and Colding.